Let's solve each equation step by step.

### 1. \( \frac{1}{\sqrt{x^2}} = 3 \)

First, simplify the left side:
\[ \frac{1}{|x|} = 3 \]

Since \( |x| \) denotes the absolute value of \( x \), we can write:
\[ |x| = \frac{1}{3} \]

This gives us two possible solutions:
\[ x = \frac{1}{3} \quad \text{or} \quad x = -\frac{1}{3} \]

### 2. \( \sqrt{16x^2} = 1 \)

Simplify the left side:
\[ 4|x| = 1 \]

Solve for \( |x| \):
\[ |x| = \frac{1}{4} \]

This gives us two possible solutions:
\[ x = \frac{1}{4} \quad \text{or} \quad x = -\frac{1}{4} \]

### 3. \( \sqrt{25x^2} - 125 = 0 \)

First, isolate the square root term:
\[ \sqrt{25x^2} = 125 \]

Simplify the left side:
\[ 5|x| = 125 \]

Solve for \( |x| \):
\[ |x| = 25 \]

This gives us two possible solutions:
\[ x = 25 \quad \text{or} \quad x = -25 \]

### 4. \( \sqrt{36x^2} = |-12| \)

First, simplify the right side:
\[ |-12| = 12 \]

Now, simplify the left side:
\[ 6|x| = 12 \]

Solve for \( |x| \):
\[ |x| = 2 \]

This gives us two possible solutions:
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Summary of Solutions

1. \( x = \frac{1}{3} \) or \( x = -\frac{1}{3} \)
2. \( x = \frac{1}{4} \) or \( x = -\frac{1}{4} \)
3. \( x = 25 \) or \( x = -25 \)
4. \( x = 2 \) or \( x = -2 \)